Two-point Functions in a Holographic Kondo Model

We develop the formalism of holographic renormalization to compute two-point functions in a holographic Kondo model. The model describes a \((0+1)\)-dimensional impurity spin of a gauged \(SU(N)\) interacting with a \((1+1)\)-dimensional, large-\(N\), strongly-coupled Conformal Field Theory (CFT). We describe the impurity using Abrikosov pseudo-fermions, and define an \(SU(N)\)-invariant scalar operator \(\mathcal{O}\) built from a pseudo-fermion and a CFT fermion. At large \(N\) the Kondo interaction is of the form \(\mathcal{O}^{\dagger} \mathcal{O}\), which is marginally relevant, and generates a Renormalization Group (RG) flow at the impurity. A second-order mean-field phase transition occurs in which \(\mathcal{O}\) condenses below a critical temperature, leading to the Kondo effect, including screening of the impurity. Via holography, the phase transition is dual to holographic superconductivity in \((1+1)\)-dimensional Anti-de Sitter space. At all temperatures, spectral functions of \(\mathcal{O}\) exhibit a Fano resonance, characteristic of a continuum of states interacting with an isolated resonance. In contrast to Fano resonances observed for example in quantum dots, our continuum and resonance arise from a \((0+1)\)-dimensional UV fixed point and RG flow, respectively. In the low-temperature phase, the resonance comes from a pole in the Green's function of the form \(-i \langle {\cal O} \rangle^2\), which is characteristic of a Kondo resonance.